Fuzzy scalar field theory as matrix quantum mechanics

arX

iv:1

012.

3568

v2 [

hep-

th]

20

Mar

201

1

HWM–10–36

EMPG–10–26

ZMP–HH/10–394

Fuzzy Scalar Field Theory

as Matrix Quantum Mechanics

Matthias Ihl∗, Christoph Sachse† and Christian Samann‡

∗ Instituto de Fısica

Universidade Federal do Rio de Janeiro

21941-972 Rio de Janeiro, RJ, Brasil

Email: [email protected]

† Fachbereich Mathematik

Bereich Algebra und Zahlentheorie

Universitat Hamburg

D-20146 Hamburg, Deutschland

Email: [email protected]

‡ Department of Mathematics

Heriot-Watt University

Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K.

and Maxwell Institute for Mathematical Sciences, Edinburgh, U.K.

Email: [email protected]

Abstract

We study the phase diagram of scalar field theory on a three dimensional Eu-

clidean spacetime whose spatial component is a fuzzy sphere. The corresponding

model is an ordinary one-dimensional matrix model deformed by terms involving

fixed external matrices. These terms can be approximated by multitrace expres-

sions using a group theoretical method developed recently. The resulting matrix

model is accessible to the standard techniques of matrix quantum mechanics.

1. Introduction and results

Fuzzy spaces, such as the fuzzy sphere S2F [1], provide an interesting way of regularizing

quantum field theories [2, 3]: These spaces come with an algebra of functions which is finite

dimensional. Correspondingly, the functional integral appearing in the partition function of a

quantum field theory on a fuzzy space reduces to a finite dimensional integral. An advantage

of this approach over a lattice regularization is that it preserves symmetries: the isometries

of the classical manifold still have a well-defined action on the corresponding fuzzy space.

Interestingly, (scalar) quantum field theories on a fuzzy sphere1 correspond to hermitian

matrix models with additional couplings to fixed external matrices [4, 5, 6]. These matrices

originate from the kinetic term in the action, and they yield an obstruction to applying

the usual matrix model technology for reducing the partition function to an integral over

eigenvalues. Nevertheless, one can use group theoretical methods to perform this reduction

after Taylor-expanding the exponential of the kinetic term in the partition function [5, 6].

The rewritten partition function can then be used to derive analytically the phase structure

of scalar field theory on the fuzzy sphere and to compare the result to the numerical studies

of [7, 8, 9, 10, 11, 12]: Qualitatively, the phase structures match and the position of a triple

point appearing in the phase diagram roughly agreed in both the numerical and analytical

results.

In this paper, we continue the work of [5, 6] by studying scalar field theories on the

three-dimensional spacetime R × S2F . The resulting model is a special variant of matrix

quantum mechanics with additional couplings to fixed external matrices. We can use again

the group theoretical techniques developed in [5]. That is, we Taylor-expand the exponential

of the spatial part of the kinetic term in the partition function and rewrite the resulting

series order by order in terms of multitrace expressions. Re-exponentiating these yields the

partition function of a matrix quantum mechanical model involving the multitrace terms. In

fact, we can trivially translate the results obtained in [6] to the matrix quantum mechanics

model.

Similar to a pure hermitian matrix model, matrix quantum mechanics can be solved in

the large N limit. One approach is to map the system to a non-interacting Fermi gas [13], an

alternative way is the collective field theory formalism developed in [14]. These techniques

can still be applied if the potential contains multitrace terms of the matrix field, as done e.g.

in [15, 16, 17] in the 2d gravity context. The model that we obtain will be more general in

the deformations than the models described in these papers, but it will still be accessible to

these techniques.

The purpose of this paper is to complement the numerical results on the phase diagram of

scalar φ4-theory on R×S2F found in [18, 19, 20] by an analytical study. Using the techniques

mentioned above, we are able to derive many properties of this phase diagram. In particular,

we confirm the existence of a third phase as compared to pure matrix quantum mechanics.

Furthermore, we find expressions for the various phase boundaries, and in an important

region of the parameter space, we can write down explicit formulas for the free energy. In

the remaining parameter space, we can calculate the free energy for each point using two

simple numerical operations.

1or, more generally, fuzzy projective algebraic varieties

1

This paper is structured as follows. In section 2, we briefly review scalar quantum field

theories on fuzzy complex projective spaces and show how to turn these into matrix quantum

mechanics. An expression for the free energy of such a theory on a fuzzy sphere is derived in

section 3. In section 4, we discuss general features of the phase diagram of this theory and

quantitative results are presented in section 5.

2. Fuzzy scalar field theory as multitrace matrix quantum mechanics

2.1. Fuzzy complex projective spaces

We will work with Berezin-quantized complex projective space as described e.g. in [6] and

we briefly summarize the associated notions in the following. Consider the space Cn+1 with

complex coordinates zα, α = 0, ..., n. Using the fibration U(1) → S2n+1 → CPn, we conclude

that the set Σℓ of functions on Cn+1 of the form

f(z) =∑

αi,βi

fα1...αℓβ1...βℓzα1

...zαℓzβ1

...zβℓ

|z|2ℓ , fα1...αℓβ1...βℓ ∈ C (2.1)

descends to a subset of the smooth functions on CPn ⊂ S2n+1 ⊂ Cn+1. The Hilbert space

Hℓ for Berezin-quantized CPn is the ℓ-particle Hilbert space in the Fock space of n + 1

harmonic oscillators. For functions in Σℓ of the form (2.1), the quantization prescription is

explicitly given by

f(z) 7→ f =∑

αi,βi

fα1...αℓβ1...βℓ1

ℓ!a†α1

...a†αℓ|0〉〈0|aβ1

...aβℓ. (2.2)

Note that real functions f are mapped to hermitian operators f ∈ End (Hℓ) and the constant

function f(z) = c, c ∈ R, is mapped to c · 1 ∈ End (Hℓ). Furthermore, Nn,ℓ := dim(Hℓ) =(n+ℓ)!n!ℓ! and a real function f ∈ Σℓ can be interpreted as an Nn,ℓ×Nn,ℓ-dimensional, hermitian

matrix.

The Laplace operator ∆ on Σ can be lifted to End (Hℓ) and the lift is given by the

quadratic Casimir C2 of SU(n+1) in the representation formed by End (Hℓ). One can show

that [21]

∆f = C2f . (2.3)

Furthermore, we can define an integral operation on End (Hℓ) by taking the trace:

∫dµFS f =

vol(CPn)

Nn,ℓtr (f) , (2.4)

where dµFS is the Liouville measure induced by the Fubini-Study metric on CPn. For a

more detailed exposition of these relations, see e.g. [21].

2.2. Partition function of the model

The action of scalar field theory with quartic potential on R×CPnF is given by

S[Φ] = β

∫dt tr

(12Φ(t)

(C2 − ∂2

t

)Φ(t) + rΦ2(t) + gΦ4(t)

). (2.5)

2

The scalar fields are represented by the time-dependent, Nn,ℓ ×Nn,ℓ-dimensional hermitian

matrices Φ(t) ∈ C∞(R,End (Hℓ)) and, as mentioned above, the spatial part of the Laplace

operator ∆ is given by the quadratic Casimir C2 of SU(n + 1). Note that we are working

with Euclidean time, which implies a different sign in front of the potential compared to

Minkowski signature.

Explicitly, we have C2Φ(t) := [Li, [Li,Φ]], where the Li form generators of SU(n + 1)

acting on Hℓ. We follow the conventions of [6] and normalize the generators according to

[Li, Lj ] =: i∑

k

fijkLk , tr (Li) = 0 ,∑

i

L2i = cL1 and tr (LiLj) =

cLNn,ℓ

(n+ 1)2 − 1δij .

This yields positive eigenvalues for C2, and thus the action (2.5) is bounded from below for

g > 0.

The model (2.5) defines a one-dimensional field theory of matrix-valued scalar fields.

Such theories are known as matrix quantum mechanics in the literature. The corresponding

partition function reads

Z =

∫DΦ(t) exp (−βS[Φ]) , (2.6)

where DΦ(t) is the measure over functions on R taking values in the Lie algebra of hermitian

matrices of size Nn,ℓ × Nn,ℓ. For each value of t, the measure corresponds to the Dyson

measure2 on the space of hermitian matrices. As in the case of the pure matrix model, we

can therefore split the partition function into an eigenvalue part DΛ(t) and an angular part

DΩ(t):

Z =

∫DΛ(t)DΩ(t) exp (−βS[Φ]) . (2.7)

One now usually exploits the fact that expressions in the action which consist exclusively of

traces of polynomials in the Φ are independent of the angular part: Diagonalizing Φ according

to Λ := Ω†ΦΩ, where Λ is the diagonal matrix of eigenvalues of Φ and Ω is some unitary

matrix, yields the simplification

tr (Φn) = tr ((ΩΦΩ†)n) = tr (Λn) . (2.8)

The fixed external matrices Li originating from the quadratic Casimir, however, present an

obstacle to rewriting the action in terms of Λ. In [5] the same problem was analyzed for scalar

field theory on fuzzy spaces, that is, the dimensional reduction of the model (2.5). There it

was suggested to perform a Taylor expansion of the exponential of the kinetic term in the

partition function. Order by order, the terms in this expansion can be evaluated analytically

using group theoretical methods and, when recombined, yield the partition function of a

matrix model with an action containing multitrace terms. We can apply the same method

here and directly take over the results obtained in [6]: The Lagrangian of our model is given

by

L = tr(− Φ∂2

tΦ+ rΦ2 + gΦ4)+ LCasimir , (2.9)

2i.e. the standard translation-invariant measure induced by the bi-invariant Haar measure on U(Nn,ℓ)

3

where LCasimir = tr (Φ(t)C2Φ(t)). For the fuzzy sphere CP 1F , the latter term can be approx-

imated in the limit of large matrix sizes by

LCasimir =

(− 1

Ntr (Φ)− β

3N3tr (Φ)3

)tr (Φ) +

(1− β

3Ntr (Φ2) +

2β

N2tr (Φ)2

)tr (Φ2) ,

(2.10)

where we abbreviated N = N1,ℓ. We thus arrive at a standard one-dimensional matrix

model, deformed by the multitrace terms contained in LCasimir. The corresponding actions

for CPnF with n > 1 are easily deduced from the results in [6], too: They merely correspond

to replacing the coefficients in (2.10) by more complicated expressions in N . Since our goal

is primarily to reproduce the numerical results of [18, 19], we will focus our attention on the

case of the fuzzy sphere in the following.

3. Evaluation of the free energy

In this section, we use collective field theory [14] to compute the free energy of our one-

dimensional matrix model. To leading order in the matrix size, this method is equivalent to

the semi-classical approximation of the model as presented in [13], see also [22] for a nice

review.

3.1. The free energy of multitrace matrix quantum mechanics

We start from the general one-dimensional matrix model with Lagrangian

L = 12 tr (Φ

2) + u2 tr (Φ2) + u4 tr (Φ

4)

+ v12 tr (Φ)2 + v14 tr (Φ)

4 + v22 tr (Φ2)2 + w12 tr (Φ)

2 tr (Φ2)

= 12 tr (Φ

2) + V .

(3.1)

For simplicity, we decompose the matrix Φ into Φ =∑

aΦaτa, where τa are hermitian

generators of u(N), normalized according to tr (τaτb) = δab. The Hamiltonian corresponding

to (3.1) can be written as

H = −12

∑

a

∂2

∂Φ2a

+u2 tr (Φ2) + u4 tr (Φ

4)

+v12 tr (Φ)2 + v14 tr (Φ)

4 + v22 tr (Φ2)2 + w12 tr (Φ)

2 tr (Φ2) .

(3.2)

We now switch to the collective fields φ(λ) ∈ Cc(R) via the Fourier transform

φ(λ) =

∫dk

2πNeikλ tr (e−ikΦ) , (3.3)

where we inserted a factor of 1N compared to [14] to facilitate the large N limit. Here, the

collective field φ(λ) will turn out to play a similar role as the eigenvalue density in the case

of the hermitian matrix model. We correspondingly introduce the various moments of φ(λ):

ck :=

∫dλφ(λ)λk . (3.4)

4

Note that (3.3) implies that the collective field φ(λ) satisfies the normalization condition

c0 =

∫dλφ(λ) =

1

Ntr (1) = 1 . (3.5)

The potential V , i.e. the non-derivative terms in the Lagrangian (3.1), is easily seen to

transform into

V =1

N

∫dλφ(λ)

((v12c1 + v14c

31)λ+ (u2 + v22c2 + w12c

21)λ

2 + u4λ4). (3.6)

The transformation of the kinetic term is technically more involved, see [14]. Here, let us

just note that one eventually arrives at the transformed Hamiltonian H = T + V where the

effective potential V reads as

V =

∫dλφ(λ)

(12NG2(λ, φ)

)+ V (3.7)

with the resolvent G(λ, φ) =∫dζ φ(ζ)

ζ−λ . To determine the collective field φ0(λ) for the ground

state of the system, we have to minimize the functional

E(µF , νF , κF , φ) = V + µF

(1−

∫dλφ(λ)

)

+ νF

(c1 −

∫dλφ(λ)λ

)+ κF

(c2 −

∫dλφ(λ)λ2

).

(3.8)

The variations with respect to φ, µF , νF , κF , c1 and c2 yield the system of equations

12

(∫− dζ

φ(ζ)

ζ − λ

)2

−∫− dζ

φ(ζ)

ζ − λ

∫− dξ

φ(ξ)

ξ − ζ

= µF + νFλ+ κFλ2 −

((v12c1 + v14c

31)λ+ (u2 + v22c2 + w12c

21)λ

2 + u4λ4),

1 =

∫dλφ(λ) , c1 =

∫dλφ(λ)λ , c2 =

∫dλφ(λ)λ2 ,

νF = −v12c1 − 3v14c31 − 2w12c1c2 , κF = −v22c2 ,

(3.9)

which is solved by standard methods, cf. [14, 23]. We arrive at the collective field

φ0(λ) =

1

π

√2 (µF − a1λ− a2λ2 − a4λ4) for λ ∈ I ,

0 else ,(3.10a)

where

a1 = 2v12c1 + 4v14c31 + 2w12c1c2 , a2 = u2 + 2v22c2 + w12c

21 , a4 = u4 , (3.10b)

and I ⊂ R can be any union of closed intervals such that φ0(λ) = 0 at the boundaries.

Equations (3.9) imply that

∫dλφ0(λ)G

2(λ, φ0) =13

∫dλφ0(λ)

(2(µF − a1λ− a2λ

2 − a4λ4))

=π2

3

∫dλφ3

0 , (3.11)

5

and the potential V reads as

V =1

N

∫dλφ0(λ)

(−π2

2φ20(λ) + µF − c1v12λ− c2v22λ

2 − 3c31v14λ+ 2c1c2w12λ

).

Together with (3.10), we arrive at the following expression for the energy (3.8) in the ground

state:NE(µF , νF , κF , φ0) =µF − v12c

21 − v22c

22 − 3v14c

41 + 2w12c

21c2

− 13

∫dλ

π

(2(µF − a1λ− a2λ

2 − a4λ4)) 3

2 .(3.12)

While all the integrals appearing above can be expressed in terms of elliptic functions as done

in appendix A, it is not possible to use them to find analytic expressions for µF in general.

3.2. Comments on the semiclassical approximation

The semi-classical approximation employed in [13] leads to the same result as the collective

field theory method. Here, one considers the quantum mechanical problem of finding the

eigenvalues of the N -particle Hamiltonian corresponding to the action (3.1), where Φ =

Φ† ∈ MatC(N). As in the case of the zero-dimensional matrix model, one can switch to an

eigenvalue formulation of the model. The Vandermonde determinant arising from this can

be absorbed in the quantum mechanical wavefunction, rendering it totally antisymmetric.

The ground state energy of this system is then found by using the standard description in

terms of free fermions, where the Lagrange multiplier µF of the collective field theory method

becomes the Fermi energy.

The only new ingredient here is the treatment of the multitrace terms: By linearizing e.g.

tr (Φ2) around the vacuum expectation value c2 = 〈 tr (Φ2)〉, we find the relation

( tr (Φ2))2 ≈ 2〈 tr (Φ2)〉 tr (Φ2)− 〈 tr (Φ2)〉2 , (3.13)

cf. [16]. This induces a constant shift of the free energy proportional to 〈 tr (Φ2)〉2 as well

as a doubling of the naıve contribution of the term corresponding to tr (Φ2)2. In [16], this

was justified from physics principles. In the previous section, we saw that precisely the same

modifications arise in our model from integrating out the Lagrange multiplier κF .

3.3. The large N limit

As usual for matrix models3, the large N limit is not unique, but requires fixing of the

scaling behavior of all coupling constants and the eigenvalues. While mathematically all

scalings which leave the potential positive definite are equally valid, the resulting models are

physically very different. One physical constraint one usually imposes is that in the limit

N → ∞, the coupling constants approach critical values.

Our model is supposed to provide an approximation for scalar field theory on R1 × S2,

and we expect all terms of the potential as well as the kinetic term to survive in the large N

limit without becoming dominant. This implies that the leading order of all the contributions

to the action should be of order O(N0), subleading contributions of order O(N−1) provide

3A similar problem is encountered in quantum field theory on the lattice.

6

corrections to the large N limit. An overall scaling of the action is irrelevant, as we can

define the free energy with a corresponding power of N .

To obtain a homogeneous scaling between the approximations due to the kinetic term

and the potential, we can use the scaling found in [6]:

β → N− 1

2β , λ → N− 1

4λ , r → N2r , g → N5

2 g , (3.14)

where r and g are the actual parameters of our model (2.9). In addition, we have to scale

t → 1N t to include the right scaling of the temporal part of the kinetic term. Altogether, our

model corresponds to the general model (3.1) with the following parameters:

u2 = 1 + r , u4 = g , v12 = −1 , v14 = −β

3, v22 = −β

3, w12 =

2

3β , (3.15)

and for λ ∈ I, the corresponding collective field reads as

φ0(λ) =1

π

√2

(µF −

(−2c1 −

4

3βc31 +

4

3βc1c2

)λ−

(1 + r − 2

3βc2 +

2

3βc21

)λ2 − gλ4

).

We furthermore have the relation

a1 = 2c1(r − a2) . (3.16)

4. The phase diagram

4.1. The three phases

We will exclusively deal with the situation in which r0 is negative enough for the potential to

have a local maximum. Note that in matrix form, our potential is symmetric under Φ → −Φ.

In the collective field reformulation, we have a symmetry of the potential V under λ → −λ.

Naively, one therefore expects that symmetric phases are the dominant ones. However, our

experience with the pure matrix model [6] suggests that we should also allow for a third

phase. This is further motivated by the numerical findings of [18, 19, 20], which also observe

(at least) three different phases. Depending on µF , we can distinguish three phases:

I. The disordered phase or single-cut case. Here, µF > 0 and the filling of the eigenvalues

covers the local maximum completely. There is a single interval I = [−λ1, λ1] over

which one has to integrate λ, and it is symmetric around the origin. This implies that

the first moment of the collective field vanishes: c1 = 0.

II. The non-uniform ordered phase or symmetric double-cut case. In this case, µF < 0

and the Fermi sea of eigenvalues splits up into two symmetric, disjoint pieces. That is,

there are two intervals I = [−λ2,−λ1] ∪ [λ1, λ2] with 0 ≤ λ1 ≤ λ2 as support for the

integral, which are again symmetric around the origin. We again have c1 = 0.

III. The uniform (ordered) phase or asymmetric double-cut case. As in phase II, µF < 0

and the Fermi sea is split, but this time into two asymmetric pieces. The length of the

two intervals I1 ∪ I2 = I is different. Note that true uniform ordering is only achieved

in the totally asymmetric case, in which one of the intervals has shrunk to zero size.

Here, c1 6= 0.

7

The actual phase is determined from existence conditions and the fact that the physical

system adopts the ground state with the lowest free energy.

V (λ)

λ

V (λ)

λ

V (λ)

λ

Figure 1: The three phases. The solid line describes the potential V (λ) felt by the eigenvalues.

The dashed lines mark the Fermi energy µF . In phase III, we chose a totally asymmetric

filling with I1 = ∅. The asymmetry of the potential in this phase is due to c1 6= 0.

Note that our model can always be treated as ordinary matrix quantum mechanics, where

the coefficients of the potential are dependent on the moments c1 and c2. That is, to study

our model, we can solve matrix quantum mechanics for a general potential and then derive

self-consistency conditions on the moments. For this reason, it is possible to evaluate the

exact location of the phase transition between phases I and II as well as the existence domain

of phase III in our model analytically.

4.2. The phase transition I to II

The phase transition between the single-cut phase and the double-cut phase obviously occurs

at µF = 0. In this case, the elliptic integrals can be performed explicitly. The interval I on

which φ0(λ) is supported is given by

I =

(−√

−u2−2v22c2u4

,√

−u2−2v22c2u4

), (4.1)

and the normalization condition yields

√8 (−u2 − 2v22c2)

3

2

3πu4= 1 . (4.2)

The second moment c2 is determined by the condition

c2 =

√32 (−u2 − 2v22c2)

5

2

15πu24=

(3π)2

3

5u1

3

4

, (4.3)

where we made use of (4.2). Plugging this into (4.2) yields a phase transition at

u2 = −(3π)2

3 (5u4 + 4v22)

10(u4)1

3

. (4.4)

If u2 is larger than the right-hand side, we are in the single-cut phase, while for u2 smaller

than the right-hand side, the system is in the double-cut phase.

8

4.3. Existence condition for phase II

Similarly to the pure matrix model discussed in [6], there is a region of the parameter space

which is not covered by phase I and where the self-consistency relation for c2 cannot hold true

in phase II. To see this, we compute c2 for the double cut solution of pure matrix quantum

mechanics with v12 = v14 = v22 = w12 = 0 for arbitrary parameters a2 < 0 and a4 > 0. We

then translate to our model by identifying

a2 = 1 + r − 2

3βc2

(a2, a4, µF (a2, a4)

)and a4 = u4 = g , (4.5)

where r and g are the actual parameters of our model. The region of the parameter

space in which we expect the self-consistency relation to be problematic corresponds to

0 < a4/(−a2) ≪ 1. In this region, the wells in the potential are sufficiently deep to be

approximated by a parabola. That is, we approximate e.g. the right well as follows

2(µF − a2λ2 − a4λ

4) ≈ 2µF +a222g

+ 4a2

(λ−

√−a22g

)2

. (4.6)

Both the integrals yielding the normalization condition for φ0 and the second moment c2 can

be computed and read as

∫dλφ0(λ) ≈

a22 + 4gµF

4g√−a2

,

∫dλφ0(λ)λ

2 ≈ (a22 + 4gµF )(17a22 + 4gµF )

128g2(−a2)3/2.

(4.7)

We can use these results together with (4.5) to determine the function r(a2):

r ≈ −1 +β

12√−a2

− 1

3a2

(β

g− 3

). (4.8)

Note that the larger |a2|, the better this approximation becomes. In particular, we see that

for g ≤ 13β, r grows as a2 becomes more negative. This implies that phase II exists for

arbitrarily large values of |r| only if g > 13β.

4.4. Existence condition for a totally asymmetric phase III

To simplify our analysis, we will identify the third phase with a totally asymmetric filling.

That is, I is again a single interval, filling only one of the two wells in the potential. It is

obvious that the transition from phase I to the totally asymmetric filling has to go smoothly

through all possible asymmetric fillings, as close to the boundary between phases I and II,

no totally asymmetric solution will exist.

The existence boundary for the totally asymmetric solution is another line in our phase

diagram which can be determined analytically if we neglect the contribution of the odd

moment c1. As we will see later in our numerical studies, the odd moments decrease the

depth of the filled well. The bound obtained by this approximation for the existence of a

totally asymmetric phase III is therefore an upper bound.

In our approximation, this line is determined by the fact that one of the two wells of the

potential is filled up to the local maximum and that a further increase in Fermi energy µF

9

would lead to a spilling of eigenvalues into the other well. Consequently, we put µF = 0 and

consider the interval

I =

(0,√

−u2−2v22c2u4

). (4.9)

The computation of the existence boundary then proceeds exactly as the computation in the

previous section. We obtain

√2(−u2 − 2v22c2)

3

2

3πu4= 1 and c2 =

√8 (−u2 − 2v22c2)

5

2

15πu24=

(6π)2

3

5u1

3

4

, (4.10)

and the existence domain of the totally asymmetric phase is given by

u2 ≤ −(3π)2

3 (5u4 + 4v22)

5(2u4)1

3

. (4.11)

4.5. Pure matrix quantum mechanics

Before discussing the phase diagram of our model, let us briefly discuss the slightly simpler

case of pure matrix quantum mechanics, for which v12 = v14 = v22 = w12 = 0. Here, the

phase transition between I and II occurs at

u4 = −√8(−u2)

3

2

3π, (4.12)

and the existence boundary for the totally asymmetric cut is

u4 = −√2(−u2)

3

2

3π. (4.13)

3.0

2.5

2.0

1.5

1.0

0.5

-----5 4 3 2 1

u4

u2

III

II

Figure 2: The phase diagram for pure matrix quantum mechanics. The solid line describes

the phase transition and the dashed line the boundary of the existence domain of the totally

asymmetric cut. The roman numerals describe the actual phases of the system.

10

One can now show by a simple physical argument that the lowest energy configuration is

always the symmetric double-cut solution, and thus phase III does not exist in pure matrix

quantum mechanics: In the time-independent ground state and for large N , one would

expect that moving one eigenvalue from one well to another does not cost or yield energy.

This automatically implies that the Fermi energy in both wells is the same. To be rigorous,

one can introduce filling fractions ρ1 and ρ2 for the left and right wells of the potential

with ρ1 + ρ2 = 1 and∫Iiφ0(λ) = ρi. One can then determine the true minimum of the

free energy numerically, which confirms the physical argument. The phase diagram of pure

matrix quantum mechanics is depicted in figure 2.

5. Results

In the following, we put β = 12 and perform an analysis of the phase diagram relying on solving

the system of equations (3.10) and determining the free energy. As we already computed

the location of the phase transition between phases I and II, it remains to compare the free

energy on the overlap of the existence domains of phases II and III.

5.1. Phases I and II

Recall that the filling of the wells with eigenvalues is symmetric and therefore the odd mo-

ments of the collective field, and in particular c1, vanish. Equations (3.10) cannot be solved

analytically, as they involve elliptic integrals. We therefore apply the following algorithm:

1. pick a value for a4 = g and a2

2. evaluate µF for this pair by numerically minimizing |1−∫Idλφ0(λ)|

3. numerically evaluate the second moment c2

4. deduce the value of r from a2 and c2

5. evaluate the free energy at the point (r, g) in the parameter space

By applying this algorithm to a range of values for a2 and a4, we can perform a sweep of

the r-g-parameter space. The only restriction here is the existence boundary for phase II for

g ≤ 13β.

The region of the r-g-parameter space, in which we expect an overlap between phases

II and III is characterized by large |r| and small g. The potential in this region has broad

and deep wells, and correspondingly µF is small compared to the depth of the wells. This

implies that an approximation of the potential by a parabola, like the one used in section

4.3., should work reasonably well.

Explicitly, we approximate the collective field by

φ0(λ) ≈

1

π

√p0 − α(λ− λmin)2 for λ ∈ I ,

0 else .(5.1)

The zeros of this collective field φ0(λ) defining its support I = [−λR,−λL] ∪ [λL, λR] are

λL = λmin −√

p0α

and λR = λmin +

√p0α

. (5.2)

11

Together with the integrals given in appendix B, we can now compute

p0 =√α , c2 =

1

4√α+ λ2

min , a2 = 1 + r − β

6√α− 2

3βλ2

min . (5.3)

From these relations, we derive

r = −1− α

4+

β

6√α+

2βλ2min

3, g =

α

8λ2min

,

NE =

√α

4+

β

48α− αλ2

min

8+

βλ2min

6√α

+βλ4

min

3.

(5.4)

Rewriting the free energy NE in terms of r and g yields a lengthy but analytic expression,

which is plotted in figure 3. Note that one can easily check the accuracy of the approximation

by verifying the exact normalization condition as well as the self-consistency condition for c2using the approximate values of µF and c2. This plot is confirmed by the numerical results

of the first algorithm.

200

400

10

20

g1.0

0.5

-

-

-

-

0

r

Figure 3: The free energy in phase II as obtained from the approximation of the potential

wells by parabolas.

5.2. Phase III

Phase III is more intricate to deal with due to the additional appearance of the terms involving

the odd moment c1. From the expression of the free energy (3.12), it is clear that the free

energy in phase III is much larger than the free energy in phase II if c2 and the value of

the integral are assumed to be roughly the same. We therefore do not expect phase III to

compete with phase II, but to exist only where phase II does not. This is precisely the case

in the region of the r-g-parameter space for which the approximation of the wells of the

potential by parabolas, and thus (5.1), works well.

For a totally asymmetric filling, i.e. a collective field φ0(λ) with support I = [λL, λR] we

obtain

p0 = 2√α , c1 = λmin , c2 =

1

2√α+ λ2

min , a2 = 1 + r − β

3√α

. (5.5)

12

This leads to the following expressions:

r =1

12

(6− 3α− 2β√

α

), g =

α− 2

8λ2min

+β

12√αλ2

min

,

NE =12α3/2 + 2β + 6αλ2

min − 3α2λ2min + 30

√αβλ2

min + 64αβλ4min

24α.

(5.6)

A plot of the free energy in phase III according to this approximation is given in figure 4.

200

400

6001020

1.51.0

0.5

--

0

r

g

Figure 4: The free energy in phase III as obtained from the approximation of the potential

wells by parabolas.

5.3. Discussion

In the region of the parameter space in which phases II and III coexist, the free energy is

negative in phase II, while it is positive in phase III. Note that this observation is due to

our restriction of phase III to a totally antisymmetric eigenvalue filling. Actually, one would

expect the system to adopt intermediate phases so that the free energy changes continuously

between the extrema of a symmetric and a totally asymmetric filling.

From the different signs of the free energy in phases II and III it follows that phase

III is only adopted by the system if phase II is not available. For large |r|, we therefore

expect a phase transition at the line g = 13β, below which phase II does not necessarily exist.

Interestingly, the same linear phase boundary between phases II and III also appeared in the

numerical results of [18]. The fact that our phase boundary only holds for large |r| is due

to our approximation of the exponential of the kinetic term of the original model to second

order. We expect higher orders to provide the necessary further corrections. Also, higher

order contributions will yield terms in the potential of the one-dimensional matrix model

which are of order tr (Φn) with n > 4. These terms would give rise to more than two wells

in the potential leading to additional phases.

In summary, we conclude that the phase diagram of our theory is a slight deformation of

that of matrix quantum mechanics, with the addition of a third phase. The phase boundaries

between phases I and II and phases II and III are given, respectively, by the two curves

r = −(3π)2

3 (5g − 4)

10g1

3

and g =1

3β . (5.7)

13

Acknowledgements

C. Samann would like to thank Denjoe O’Connor for discussions. The work of M. Ihl was

supported by a grant from CNPq (Brazilian funding agency). C. Sachse was partially sup-

ported by a postdoctoral research fellowship of the Deutsche Forschungsgemeinschaft while

this work was completed. The work of C. Samann was supported by a Career Acceleration

Fellowship from the UK Engineering and Physical Sciences Research Council.

Appendix

A. The elliptic integrals

Throughout the paper, we encountered elliptic integrals, which can be reduced to the follow-

ing ones:

I1 =∫ b

adλ

√(b2 − λ2)(λ2 − a2) , J1 =

∫ b

0dλ

√(b2 − λ2)(λ2 + a2) ,

I2 =∫ b

adλλ2

√(b2 − λ2)(λ2 − a2) , J2 =

∫ b

0dλλ2

√(b2 − λ2)(λ2 + a2) ,

I3 =∫ b

adλ

(√(b2 − λ2)(λ2 − a2)

)3, J3 =

∫ b

0dλ

(√(b2 − λ2)(λ2 + a2)

)3,

(A.1)

with b > a ≥ 0. These integrals can be computed explicitly in terms of elliptic functions. We

have4

I1 =i

3b((a2 + b2

)E0 −

(a2 + b2

)E2 −

(a2 − b2

)(F2 −K0)

), (A.2a)

J1 =1

3

(a(−a2 + b2

)E1 + a

(a2 + b2

)K1

), (A.2b)

I2 =i

15b(2(a4 − a2b2 + b4

)E0 − 2

(a4 − a2b2 + b4

)E2

+(a4 − 3a2b2 + 2b4

)(F2 −K0)

),

(A.2c)

J2 =1

15a(2(a4 + a2b2 + b4

)E1 −

(2a4 + 3a2b2 + b4

)K1

), (A.2d)

I3 =i

35b(2(a6 − 5a4b2 − 5a2b4 + b6

)E0 − 2

(a6 − 5a4b2 − 5a2b4 + b6

)E2

+(a6 + 8a4b2 − 11a2b4 + 2b6

)(F2 −K0)

),

(A.2e)

J3 =1

35

(2a

(−a6 − 5a4b2 + 5a2b4 + b6

)E1 + a

(a2 + b2

) (2a4 + 9a2b2 − b4

)K1

), (A.2f)

4Our conventions for elliptic functions agree with those of Mathematica.

14

where we abbreviated complete and incomplete elliptical functions as follows:

E0 := E

(a2

b2

), E1 := E

(− b2

a2

), E2 := E

(arcsin

(b

a

) ∣∣∣ a2

b2

),

F2 := F

(arcsin

(b

a

) ∣∣∣ a2

b2

),

K0 := K

(a2

b2

), K1 := K

(− b2

a2

).

(A.3)

B. Further integrals

The integrals given below are used in the analysis of phase III. Recall our approximation for

the collective field:

φ0(λ) =

1

π

√p0 − α(λ− λmin)2 for λ ∈ I ,

0 else .(B.1)

The zeros of this collective field define I = [λL, λR] and are located at

λL = λmin −√

p0α

and λR = λmin +

√p0α

. (B.2)

Assuming that the expression under the square root in φ0 is positive for λ ∈ [λL, λR], we

have: ∫ λR

λL

dλφ0(λ) =p0

2√α

,

∫ λR

λL

dλφ0(λ)(p0 − α(λ− λmin)2)2 =

3p208√α

,

∫ λR

λL

dλφ0(λ)λ =p0λmin

2√α

,

∫ λR

λL

dλφ0(λ)λ2 =

p0(p0 + 4αλ2min))

8α3/2.

(B.3)

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